2.6 Evolution of normal and millisecond pulsars

A simplified version of the presently favoured model [37, 95, 285, 2] to explain the formation of the
various types of systems observed is shown in Figure 7. Starting with a binary star system, a neutron star is
formed during the supernova explosion of the initially more massive star. From the virial theorem,
in the absence of any other factors, the binary will be disrupted if more than half the total
pre-supernova mass is ejected from the system during the (assumed symmetric) explosion [120, 33]. In
practice, the fraction of surviving binaries is also affected by the magnitude and direction of
any impulsive “kick” velocity the neutron star receives at birth from a slightly asymmetric
explosion [120, 20]. Binaries that disrupt produce a high-velocity isolated neutron star and an OB runaway
star [38]. The high probability of disruption explains qualitatively why so few normal pulsars have
companions. There are currently four known normal radio pulsars with massive main sequence
companions in eccentric orbits which are examples of binary systems which survived the supernova
explosion [140, 153, 295, 296, 193]. Over the next after the explosion, the neutron star may be
observable as a normal radio pulsar spinning down to a period several seconds. After this time, the
energy output of the star diminishes to a point where it no longer produces significant amounts of radio
emission.

For those few binaries that remain bound, and in which the companion is sufficiently massive to
evolve into a giant and overflow its Roche lobe, the old spun-down neutron star can gain a
new lease of life as a pulsar by accreting matter and angular momentum at the expense of the
orbital angular momentum of the binary system [2]. The term “recycled pulsar” is often used to
describe such objects. During this accretion phase, the X-rays produced by the liberation of
gravitational energy of the infalling matter onto the neutron star mean that such a system is
expected to be visible as an X-ray binary. Two classes of X-ray binaries relevant to binary and
millisecond pulsars exist: neutron stars with high-mass or low-mass companions. Detailed reviews
of the X-ray binary population, including systems likely to contain black holes, can be found
elsewhere [33].

Figure 8:

Eccentricity versus orbital period for a sample of 21 low-mass binary pulsars which are notin globular clusters, with the triangles denoting three recently discovered systems[294]. The solid lineshows the median of the predicted relationship between orbital period and eccentricity[249]. Dashedlines show 95% the confidence limit about this relationship. The dotted line shows. Figureprovided by Ingrid Stairs[294]using an adaptation of the orbital period-eccentricity relationshiptabulated by Fernando Camilo.

In a high-mass X-ray binary, the companion is massive enough that it might also explode as a
supernova, producing a second neutron star. If the binary system is lucky enough to survive the explosion,
the result is a double neutron star binary. At least five and probably eight such systems are now
known, the original example being PSR B1913+16 [129] - a radio pulsar which orbits
its companion every [312, 313]. In this formation scenario, PSR B1913+16 is an
example of the older, first-born, neutron star that has subsequently accreted matter from its
companion.

For many years, no clear example was known where the second-born neutron star was observed as a
pulsar. The discovery of the double pulsar J07373039 [44, 198], where a recycled pulsar “A”
orbits a normal pulsar “B” every , has now provided a dramatic confirmation of this
evolutionary model in which we identify A and B as the first and second-born neutron stars
respectively. Just how many more observable double pulsar systems exist in our Galaxy is not
clear. Although the population of double neutron star systems in general is reasonably well
understood (see Section 3.4.1), given that the lifetime of the second born pulsar is less than
one tenth that of the recycled pulsar, and that its radio beam is likely to be much smaller
(see Section 3.2.3), the prospects of ever finding more than a few 0737-like systems are rather
low.

Figure 9:

Companion mass versus orbital period for binary pulsars showing the whole sample where,in the absence of mass determinations, statistical arguments based on a random distribution of orbitalinclination angles (see Section4.4) have been used to constrain the masses as shown (Panel a), andonly those with well determined companion masses (Panel b). The dashed lines show the uncertaintiesin the predicted relation[307]. This relationship indicates that as these systems finished a period ofstable mass transfer due to Roche-lobe overflow, the size and hence period of the orbit was determinedby the mass of the evolved secondary star. Figure provided by Marten van Kerkwijk[330].

The companion in a low-mass X-ray binary evolves and transfers matter onto the neutron star on a
much longer time-scale, spinning it up to periods as short as a few ms [2]. Tidal forces during the accretion
process serve to circularize the orbit. At the end of the spin-up phase, the secondary sheds its outer layers
to become a white dwarf in orbit around a rapidly spinning millisecond pulsar. This model has
gained strong support in recent years from the discoveries of quasi-periodic kHz oscillations in a
number of low-mass X-ray binaries [338], as well as Doppler-shifted X-ray pulsations
from the transient X-ray burster SAX J1808.43658 [340, 59]. Six other “X-ray millisecond
pulsars” are now known with spin rates and orbital periods ranging between and
respectively [339, 220, 144]. Despite intensive searches [43], no radio pulsations
have so far been detected in these binaries. This could be a result of free-free absorption of
any radio waves by the thick accretion disk, or perhaps quenching the accelerating potential
in the neutron star magnetosphere by infalling matter. More sensitive radio observations are
ultimately required to place stringent limits on the physical processes responsible for the lack of
emission.

Numerous examples of these systems in their post X-ray phase are now seen as the millisecond
pulsar-white dwarf binary systems. Presently, 20 of these systems have compelling optical identifications of
the white dwarf companion, and upper limits or tentative detections have been found in about 30
others [330]. Comparisons between the cooling ages of the white dwarfs and the millisecond pulsars confirm
the age of these systems and suggest that the accretion rate during the spin-up phase was well below the
Eddington limit [114].

Further support for the above evolutionary scenarios comes from two correlations in the observed
sample of low-mass binary pulsars. Firstly, as seen in Figure 8, there is a strong correlation
between orbital period and eccentricity. The data are very good agreement with a theoretical
relationship which predicts a relic orbital eccentricity due to convective eddy currents in the
accretion process [249]. Secondly, as shown in Panel b of Figure 9, where companion masses
have been measured accurately, through radio timing (see Section 4.4) and/or through optical
observations [330], they are in good agreement with a relation between companion mass and
orbital period predicted by binary evolution theory [307]. A word of caution is required in
using these models to make predictions, however. When confronted with a larger ensemble of
binary pulsars using statistical arguments to constrain the companion masses (see Panel a of
Figure 9), current models have problems in explaining the full range of orbital periods on this
diagram [294].

The range of white dwarf masses observed is becoming broader. Since this article originally
appeared [179], the number of “intermediate-mass binary pulsars” [49] has grown significantly [54]. These
systems are distinct to the millisecond pulsar-white dwarf binaries in several ways:

The spin period of the radio pulsar is generally longer ().

The mass of the white dwarf is larger (typically ).

The orbit, while still essentially circular, is often significantly more eccentric ().

They do not necessarily follow the mass-period or eccentricity-period relationships.

It is not presently clear whether these systems originated from low- or high-mass X-ray binaries. It was
suggested by van den Heuvel [329] that they have more in common with high-mass systems. More recently,
it has been proposed [172] that a thermal-viscous instability in the accretion disk of a low-mass
X-ray binary could truncate the accretion phase and produce a more slowly spinning neutron
star.